THE DIVINE PROPORTION

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold, the second we may name a precious jewel.
--Johannes Kepler

In a paragraph towards the end of his 1611 essay On the Six Cornered Snowflake Kepler mentions the “Divine Proportion" (Golden section) and the "fibonacci sequence” in practically the same breath as flowers and pentagons.

"It is in the likeness of this self-developing series that the faculty of propagation is, in my opinion, manifest: and so in a flower the authentic flag of this faculty is shown, the pentagon." (emphasis added)

What does Kepler mean here?

First, we examine the nature of the logarithmic spiral, for reasons that will become evident below.

The 17th century mathematician Jakob Bernoulli named the figure at right the Spira mirabilis or "miraculous spiral" and assigned it the following motto: "Eadem mutato resurgo" ("although changed, I rise again the same").

The logarithmic spiral does not change its shape as its size increases. This feature is known as self-similarity. If we could zoom into the coils of the figure near the origin and enlarge them, they would fit precisely on the larger spiral.

The spiral has another extraordinary property: turning by equal angles increases the distance from the pole to the spiral by equal ratios.

What are fibonacci numbers?

In the "fibonacci sequence," referenced by Kepler, each number is the sum of the two proceeding numbers (1, 2, 3, 5, 8, 13, 21...). Therefore, the sequence can be called a "self-developing" series.

Interestingly, dividing two adjacent fibonacci number (8/5 or 21/13, for example) by each other produces increasingly precise approximation of the "Divine Proportion," which we will explore below, as the numbers grow larger.

Why does Kepler mention flowers?

Scientists, beginning with Leonardo da Vinci, observed that the displacement of leaves around a stem occurs in patterns defined by the fibonacci series.

This phenomenon is known as "philotaxis."

In the case of roses an angle that is congruent to 360 degrees by the Divine Proportion (137.5 degrees) separates the petals from each other.

In other plants the number of leaves measured around a screw-type displacement between one leaf and the leaf directly above it, and the number of rows separating the two are both fibonacci numbers.

The same phenomenon occurs in pinecones and the hearts of sunflowers.

How is the Divine Proportion related to the pentagon?

The Divine Proportion plays an essential role in the construction of the pentagon and five-fold symmetry.

To see this, examine the diagonals in a pentagon. The ratio of the diagonal to the side of a pentagon is the Divine Proportion. Moreover, the diagonals create an isosceles triangle (where two of the three sides are equal) with angles of 72 degrees and 36 degrees. This triangle can be reproduced inside itself to infinity (in a "self-developing" manner), as shown below.

In constructions of tiled pentagons, every segment is smaller than its predecessor by a factor equal to the Divine Proportion.

(the ratio of "a" to "b" is the Divine Proportion, the ratio of "b" to "c" is the Divine Proportion, and so on.

How does this relate to the logarithmic spiral?

In a rectangle where the ratio of the larger side to the smaller one is the Divine Proportion, the ratio of the sides of the "daughter rectangles" will still conform to the Divine Proportion when squares are cut from the original rectangle. Connecting the points where this series of "whirling squares" divides the sides will generate a logarithmic spiral that coils inward.

Similarly, connecting the vertices of the nested "Golden Triangles" inside the pentagon will produce the logarithmic spiral as well.

What is the "extreme and mean ratio?"

Euclid defines the "extreme and mean ratio," or "Divine Proportion," as the following relationship: When a line is divided such that the smaller section of the line (BC) is related to the larger section of the line (AC) in the same ratio as the larger section is related to the whole line (AC), then the line is divided in the Divine Proportion.

The search for harmonic relationships in the natural world motivated Kepler's scientific investigations. Whether endeavoring to understand the structure of snowflakes and or seeking the principles that underlie the organization of the cosmos as a whole, Kepler saw congruencies in nature and between the human intellect and the material world as evidence of the goodness of God and signposts of man's divine potential.

Kepler wrote, "For the theater of the world is so ordered that there exist in it suitable signs by which human minds, likenesses of God, are not only invited to study the divine works, from which they may evaluate the founder's goodness, but are also assisted in inquiring more deeply." (Optics, p. 15)

Kepler's ideas about harmony, proportion, and the nature of knowledge can be more fully appreciated with reference to the work of the renaissance statesman and mathematician, Cardinal Nicholas of Cusa (1401-1464), whose influence may be glimpsed throughout the astronomer's work.

En http://www.keplersdiscovery.com/SixCornered.html puedes ver el documento completo.

Otros documentos donde ampliar la informacion

http://tympanus.net/codrops/2012/04/17/the-divine-proportion-and-web-design/

http://www.goldennumber.net/human-body/

Phi, Pi and the

Great Pyramid of Egypt at Giza

The Great Pyramid of Egypt is based on Golden Ratio proportions

There is still some debate as to whether the Great Pyramid of Giza in Egypt, built around 2560 BC, was constructed with dimensions based on phi, the golden ratio. Its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with certainty.
There is compelling evidence, however, that the design of the pyramid embodied these foundations of mathematics and geometry:

Phi, the Golden Ratio that appears throughout nature.
Pi, the circumference of a circle in relation to its diameter.
The Pythagorean Theorem – Credited by tradition to mathematician Pythagoras (about 570 – 495 BC), which can be expressed as a² + b² = c².

First, phi is the only number which has the mathematical property of its square being one more than itself:

Φ + 1 = Φ²

1.618… + 1 = 2.618…

By applying the above Pythagorean equation to this, we can construct a right triangle, of sides a, b and c, or in this case a Golden Triangle of sides √Φ, 1 and Φ, which looks like this:

Golden Ratio Triangle of Great Pyramid of Egypt

This creates a pyramid with a base width of 2 (i.e., two triangles above placed back-to-back) and a height of the square root of Phi, 1.272. The ratio of the height to the base is 0.636.
According to Wikipedia, the Great Pyramid has a base of 230.4 meters (755.9 feet) and an estimated original height of 146.5 meters (480.6 feet). This also creates a height to base ratio of 0.636, which indicates it is indeed a Golden Triangles, at least to within three significant decimal places of accuracy. If the base is indeed exactly 230.4 meters then a perfect golden ratio would have a height of 146.53567, so the difference of only 0.3567 meters appears to be just a measurement or rounding difference.

The Great Pyramid has a surface ratio to base ratio of Phi, the Golden Ratio

A pyramid based on a golden triangle would have other interesting properties. The surface area of the four sides would be a golden ratio of the surface area of the base. The area of each trianglular side is the base x height / 2, or 2 x Φ/2 or Φ. The surface area of the base is 2 x 2, or 4. So four sides is 4 x Φ / 4, or Φ for the ratio of sides to base.

The Great Pyramid also has a relationship to Pi

There is another interesting aspect of this pyramid. Construct a circle with a circumference of 8, the same as the perimeter of this pyramid with its base width of 2. Then fold the arc of the semi-circle at a right angle, as illustrated below in “Revelation of the Pyramids”. The height of the semi-circle will be the radius of the circle, which is 8/pi/2 or 1.273.

Great Pyramid of Giza showing Phi and Pi relationships

This is less than 1/10th of a percent different than the height of 1.272 computed above using the Golden Triangle. Applying this to the 146.5 meter height of the pyramid would result in a difference in height between the two methods of only 0.14 meters (5.5 inches).

Its near perfect alignment to due north shows that little was left to chance

Some say that the relationships of the Great Pyramid’s dimensions to phi and pi either do not exist or happened by chance. Would a civilization with the technological skill and knowledge to align the pyramid to within 1/15th of a degree to true north leave the dimensions of the pyramid to chance? If they didn’t intend the precise 51.83 degree angle of a golden triangle, why would they have not used another simpler angle found in divisions of a circle such as 30, 45, 54 or 60 degrees? If the dimensions of the pyramid were not based on both phi and pi, would it not be most reasonable to assume that phi was used since it is based on the visible base of the pyramid and not an invisible circle with the same circumference as that base?

Other possibilities for Phi and Pi relationships

Even if the Egyptians were using numbers that they understood to be the circumference of the circle to its diameter and the golden ratio that appeared in nature, it’s difficult to know if they truly understood the actual decimal representations of pi and phi as we understand them now. Since references to phi don’t appear in the historical record until the time of the Greeks hundreds of years later, some contend that the Egyptians did not have this knowledge and instead used integer approximations that achieved the same relationships and results in the design.
A rather amazing mathematical fact is that pi and the square root of phi can be approximated with a high degree of accuracy using simple integers. Pi can be approximated as 22/7, resulting in a repeating decimal number 3.142857142857… which is different from Pi by only 4/100′s of a percent. The square root of Phi can be approximatey by 14/11, resulting in a repeating decimal number 1.2727…, which is different from Phi by less than 6/100′s of a percent. That means that Phi can be approximated as 256/121.
The Great Pyramid could thus have been based on 22/7 or 14/11 in the geometry shown about. Even if the Egyptians only understood pi and/or phi through their integer approximations, the fact that the pyramid uses them shows that there was likely some understanding and intent of their mathematical importance in their application. It’s possible though that the pyramid dimensions could have been intended to represent only one of these numbers, either pi or phi, and the mathematics would have included the other automatically. We really don’t know with certainty how the pyramid was designed as this knowledge could have existed and then been lost. The builders of such incredible architecture may have had far greater knowledge and sophistication than we may know, and it’s possible that both pi and phi as we understand them today could have been the driving factors in the design of the pyramid.

lunes, 7 de octubre de 2013

Cuestionario sobre Infinitos

jueves, 3 de octubre de 2013

Actividades-La divina proporcion